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A205153 Least k such that n divides s(k)-s(j) for some j where s(j)=(prime(j+1) + prime(j+2))/2.

%I A205153 #6 Mar 30 2012 18:58:10
%S A205153 2,2,3,4,3,4,6,4,5,8,5,6,9,6,7,10,7,9,10,8,9,8,14,9,10,9,11,10,14,10,
%T A205153 20,14,11,16,11,12,20,12,13,22,13,16,17,14,16,14,15,16,17,15,16,15,30,
%U A205153 16,17,16,18,17,23,17,20,31,18,20,18,19,20,19,21,20,22,20,24
%N A205153 Least k such that n divides s(k)-s(j) for some j<k, where s(j)=(prime(j+1) + prime(j+2))/2.
%C A205153 See A204892 for a discussion and guide to related sequences.
%t A205153 s[n_] := s[n] = (Prime[n + 1] + Prime[n + 2])/2; z1 = 1100; z2 = 80;
%t A205153 Table[s[n], {n, 1, 30}]     (* A024675 *)
%t A205153 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205153 Table[u[m], {m, 1, z1}]     (* A204980 *)
%t A205153 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205153 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205153 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205153 Table[d[n], {n, 1, z2}]     (* A205152 *)
%t A205153 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205153 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205153 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205153 Table[k[n], {n, 1, z2}]       (* A205153 *)
%t A205153 Table[j[n], {n, 1, z2}]       (* A205154 *)
%t A205153 Table[s[k[n]], {n, 1, z2}]    (* A205372 *)
%t A205153 Table[s[j[n]], {n, 1, z2}]    (* A205373 *)
%t A205153 Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205374 *)
%t A205153 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205375 *)
%Y A205153 Cf. A024675, A204892, A205375.
%K A205153 nonn
%O A205153 1,1
%A A205153 _Clark Kimberling_, Jan 26 2012